Question: All of the 4th grade teachers and students from Gardner Bullis went on a field trip to an archaeology museum. Tickets were $$6.50$ each for teachers and $$4.00$ each for students, and the group paid $$55.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$26.00$ each for teachers and $$9.50$ each for students, and the group paid $$163.50$ in total. Find the number of teachers and students on the field trips.
Solution: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6.5x+4y = 55.5}$ ${26x+9.5y = 163.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-26x-16y = -222}$ ${26x+9.5y = 163.5}$ Add the top and bottom equations together. $ -6.5y = -58.5 $ $ y = \dfrac{-58.5}{-6.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {6.5x+4y = 55.5}$ to find $x$ ${6.5x + 4}{(9)}{= 55.5}$ $6.5x+36 = 55.5$ $6.5x = 19.5$ $x = \dfrac{19.5}{6.5}$ ${x = 3}$ You can also plug ${y = 9}$ into $ {26x+9.5y = 163.5}$ and get the same answer for $x$ ${26x + 9.5}{(9)}{= 163.5}$ ${x = 3}$ There were $3$ teachers and $9$ students on the field trips.